A mathematical formulation for volume expansions in contouring for radiotherapy planning

Context: This research describe the characteristic volume expansion of a moving target as a function of differential margins. Aim: We aimed to ascertain the volume change after giving margin for clinical and set up uncertainties including generating internal target volume (ITV) for moving target. Materials and Methods: Settings and Design – Spheres of diameter (0.5–10 cm) with differential expansion of 1–15 mm were generated using a mathematical formula. Moving targets of radius 1–5 cm were generated, and the resultant volume envelopes with incremental motion from 1 to 20 mm were obtained. All relative volume change results were fitted with mathematical functions to obtain a generalized mathematical formula. Statistical Analysis Used: None. Results: The percentage increase in volume (%ΔVp) was much more pronounced for smaller radius target. For moving target with relatively smaller radius, %ΔVp is predominant over the absolute volume change and vice versa in case of larger radius. Mathematical formulae were obtained for %ΔVp as a function of radius and expansion and for %ΔVp in ITV volume as a function of radius and tumor movement. Conclusions: This study provides an idea of volume change for various expansions for various size targets and/or moving target for different range of movements. It establishes a correlation of these volume changes with the changing target size and range of movements. Finally, a clinically useful mathematical formulation on volume expansion has been developed for rapid understanding of the consequence of volume expansion.